muHVT: Collection of functions used for vector quantization and construction of hierarchical voronoi tessellations for data analysis

Zubin Dowlaty, Shubhra Prakash, Sangeet Moy Das, Praditi Shah, Shantanu Vaidya, Somya Shambhawi

2023-05-24

1 Abstract

The muHVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below :

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D and 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension .

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks

  4. Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required

2 Compress: Vector Quantization

This package can perform vector quantization using the following algorithms -

2.1 Hierarchical Vector Quantization

2.1.1 Using k-means

  1. The k-means algorithm randomly selects k data points as initial means
  2. k clusters are formed by assigning each data point to its closest cluster mean using the Euclidean distance
  3. Virtual means for each cluster are calculated by using all datapoints contained in a cluster

The second and third steps are iterated until a predefined number of iterations is reached or the clusters converge. The runtime for the algorithm is O(n).

2.1.2 Using k-medoids

  1. The k-medoids algorithm randomly selects k data points as initial means out of the n data points as the medoids.
  2. k clusters are formed by assigning each data point to its closest medoid by using any common distance metric methods.
  3. Virtual means for each cluster are calculated by using all datapoints contained in a cluster

The second and third steps are iterated until a predefined number of iterations is reached or the clusters converge. The runtime for the algorithm is O(k * (n-k)^2) .

These algorithm divides the dataset recursively into cells using \(k-means\) or \(k-medoids\) algorithm. The maximum number of subsets are decided by setting \(n_cells\) to, say five, in order to divide the dataset into maximum of five subsets. These five subsets are further divided into five subsets(or less), resulting in a total of twenty five (5*5) subsets. The recursion terminates when the cells either contain less than three data point or a stop criterion is reached. In this case, the stop criterion is set to when the cell error exceeds the quantization threshold.

The steps for this method are as follows :

  1. Select k(number of cells), depth and quantization error threshold
  2. Perform quantization (using \(k-means\) or \(k-medoids\)) on the input dataset
  3. Calculate quantization error for each of the k cells
  4. Compare the quantization error for each cell to quantization error threshold
  5. Repeat steps 2 to 4 for each of the k cells whose quantization error is above threshold until stop criterion is reached.

The stop criterion is when the quantization error of a cell satisfies one of the below conditions

  • reaches below quantization error threshold
  • there are less than three data points in the cell
  • the user specified depth has been attained

The quantization error for a cell is defined as follows :

\[QE = \max_i(||A-F_i||_{p})\]

where

  • \(A\) is the centroid of the cell
  • \(F_i\) represents a data point in the cell
  • \(m\) is the number of points in the cell
  • \(p\) is the \(p\)-norm metric. Here \(p\) = 1 represents L1 Norm and \(p\) = 2 represents L2 Norm.

2.1.3 Quantization Error

Let us try to understand quantization error with an example.

Figure 1: The Voronoi tessellation for level 1 shown for the 5 cells with the points overlayed

Figure 1: The Voronoi tessellation for level 1 shown for the 5 cells with the points overlayed

An example of a 2 dimensional VQ is shown above.

In the above image, we can see 5 cells with each cell containing a certain number of points. The centroid for each cell is shown in blue. These centroids are also known as codewords since they represent all the points in that cell. The set of all codewords is called a codebook.

Now we want to calculate quantization error for each cell. For the sake of simplicity, let’s consider only one cell having centroid A and m data points \(F_i\) for calculating quantization error.

For each point, we calculate the distance between the point and the centroid.

\[ d = ||A - F_i||_{p} \]

In the above equation, p = 1 means L1_Norm distance whereas p = 2 means L2_Norm distance. In the package, the L1_Norm distance is chosen by default. The user can pass either L1_Norm, L2_Norm or a custom function to calculate the distance between two points in n dimensions.

\[QE = \max_i(||A-F_i||_{p})\]

Now, we take the maximum calculated distance of all m points. This gives us the furthest distance of a point in the cell from the centroid, which we refer to as Quantization Error. If the Quantization Error is higher than the given threshold, the centroid/ codevector is not a good representation for the points in the cell. Now we can perform further Vector Quantization on these points and repeat the above steps.

Please note that the user can select mean, max or any custom function to calculate the Quantization Error. The custom function takes a vector of m value (where each value is a distance between point in n dimensions and centroids) and returns a single value which is the Quantization Error for the cell.

If we select mean as the error metric, the above Quantization Error equation will look like this :

\[QE = \frac{1}{m}\sum_{i=1}^m||A-F_i||_{p}\]

3 Projection

Sammon’s projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality while attempting to preserve the structure of inter-point distances in the projection. It is particularly suited for use in exploratory data analysis and is usually considered a non-linear approach since the mapping cannot be represented as a linear combination of the original variables. The centroids are plotted in 2D after performing Sammon’s projection at every level of the tessellation.

Denoting the distance between \(i^{th}\) and \(j^{th}\) objects in the original space by \(d_{ij}^*\), and the distance between their projections by \(d_{ij}\). Sammon’s mapping aims to minimize the below error function, which is often referred to as Sammon’s stress or Sammon’s error

\[E=\frac{1}{\sum_{i<j} d_{ij}^*}\sum_{i<j}\frac{(d_{ij}^*-d_{ij})^2}{d_{ij}^*}\]

The minimization of this can be performed either by gradient descent, as proposed initially, or by other means, usually involving iterative methods. The number of iterations need to be experimentally determined and convergent solutions are not always guaranteed. Many implementations prefer to use the first Principal Components as a starting configuration.

4 Voronoi Tessellations

A Voronoi diagram is a way of dividing space into a number of regions. A set of points (called seeds, sites, or generators) is specified beforehand and for each seed, there will be a corresponding region consisting of all points within proximity of that seed. These regions are called Voronoi cells. It is complementary to Delaunay triangulation.

Tessellate: Constructing Voronoi Tesselations

In this package, we use sammons from the package MASS to project higher dimensional data to a 2D space. The function hvq called from the HVT function returns hierarchical quantized data which will be the input for construction of the tessellations. The data is then represented in 2D coordinates and the tessellations are plotted using these coordinates as centroids. We use the package deldir for this purpose. The deldir package computes the Delaunay triangulation (and hence the Dirichlet or Voronoi tessellation) of a planar point set according to the second (iterative) algorithm of Lee and Schacter. For subsequent levels, transformation is performed on the 2D coordinates to get all the points within its parent tile. Tessellations are plotted using these transformed points as centroids. The lines in the tessellations are chopped in places so that they do not protrude outside the parent polygon. This is done for all the subsequent levels.

5 Prediction

In this package, we use predictLayerHVT function to predict based on the the sets of maps ( map A, map B, map C) constructed using HVT function. For each test records, the function will assign that record to either of Layer1 or Layer2.

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset :

  1. Calculate the distance between the point and the centroid of all the cells in the first level
  2. Find the cell whose centroid has minimum distance to the point
  3. Check if the cell drills down further to form more cells
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further

6 Example Usage : Visualizing Multidimensional Data with Sammon’s Projection using Torus (Donut)

In this section, we will see how we can use the package to visualize multidimensional data by projecting them to two dimensions using Sammon’s projection

Data Understanding

First of all, let us see how to generate data for torus. We are using a library geozoo for this purpose. Geo Zoo (stands for Geometric Zoo) is a compilation of geometric objects ranging from three to 10 dimensions. Geo Zoo contains regular or well-known objects, eg cube and sphere, and some abstract objects, e.g. Boy’s surface, Torus and Hyper-Torus.

Here, we will generate a 3D torus (a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle) with 9000 points.

set.seed(240)
# Here p represents dimension of object
# n represents number of points
torus <- geozoo::torus(p = 3,n = 9000)
torus_df <- data.frame(torus$points)
torus_df1 <- torus_df %>% round(4)
colnames(torus_df) <- c("x","y","z")

Now let’s do some EDA on the data. First of all, we will see how the data looks like


Table(head(torus_df))
x y z
-2.628238 0.5655770 -0.7253285
-1.417917 -0.8902793 0.9454533
-1.030820 1.1066495 -0.8730506
1.884711 0.1894905 0.9943888
-1.950608 -2.2506838 0.2070521
-1.482371 0.9228529 0.9672467

Now let’s have a look at structure and summary of the data.

str(torus_df)
#> 'data.frame':    9000 obs. of  3 variables:
#>  $ x: num  -2.63 -1.42 -1.03 1.88 -1.95 ...
#>  $ y: num  0.566 -0.89 1.107 0.189 -2.251 ...
#>  $ z: num  -0.725 0.945 -0.873 0.994 0.207 ...
summary(torus_df)
#>        x                  y                   z             
#>  Min.   :-2.99767   Min.   :-2.999343   Min.   :-0.9999999  
#>  1st Qu.:-1.15065   1st Qu.:-1.120632   1st Qu.:-0.7130951  
#>  Median :-0.01899   Median : 0.001856   Median : 0.0033675  
#>  Mean   :-0.00914   Mean   : 0.004195   Mean   : 0.0001237  
#>  3rd Qu.: 1.13001   3rd Qu.: 1.130708   3rd Qu.: 0.7138584  
#>  Max.   : 2.99713   Max.   : 2.999308   Max.   : 1.0000000

Now let’s try to visualize the torus (donut) in 3D Space.

library(rgl)
plotids <- plot3d(torus_df$x, torus_df$y, torus_df$z,
                  type = "s", col = c("white", "red"),
                  xlab = "X", ylab = "Y", zlab = "Z",
                  xlim = c(-5, 5), ylim = c(-5, 5), zlim = c(-5, 5))

rglwidget(elementId = "plot3drgl")

6.1 Execution of Workflow Steps at Level 1

In this section all the outlined workflow steps provided in the abstract section (Compression, Projection and Tessellation) are executed at level 1.

Step 1: Data Compression

The core function for compression in the workflow is HVQ, which is called within the HVT function. we have a parameter called quantization error. This parameter acts as a threshold and determines the number of levels in the hierarchy. It means that, if there are ‘n’ number of levels in the hierarchy, then all the clusters formed till this level will have quantization error equal or greater than the threshold quantization error. The user can define the number of clusters in the first level of hierarchy and then each cluster in first level is sub-divided into the same number of clusters as there are in the first level. This process continues and each group is divided into smaller clusters as long as thethreshold quantization error is met. The output of this technique will be hierarchically arranged vector quantized data.

However, let’s try to comprehend the HVT function first before moving on.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  projection.scale,
  normalize = T,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  diagnose = TRUE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of HVT function have been explained below :

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce reduces the level of detail captured in the data representation

We will pass the below mentioned model parameters along with torus dataset to HVT function.

Model Parameters

set.seed(240)
hvt.torus <- muHVT::HVT(
  torus_df,
  n_cells = 100,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary .

compressionSummaryTable(hvt.torus[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 100 0 0 n_cells: 100 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, none of the 100 cells have hit the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

set.seed(240)
hvt.torus2 <- muHVT::HVT(
  torus_df,
  n_cells = 300,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary again.

compressionSummaryTable(hvt.torus2[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 300 43 0.14 n_cells: 300 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

It can be observed from the table above that only 43 cells out of 300 i.e. 14% of the cells hit the Quantization Error threshold. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if 80% compression is reached

Now let’s try again with the below mentioned set of model parameters.

Model Parameters

set.seed(240)
hvt.torus3 <- muHVT::HVT(
  torus_df,
  n_cells = 900,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s check the compression summary for torus.

compressionSummaryTable(hvt.torus3[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 900 839 0.93 n_cells: 900 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

By increasing the number of cells to 900, we were successfully able to compress 93% of the data, so we will not further subdivide the cells

Step 2: Data Projection

The function sammonsProjection() utilizes the sammons function from the MASS package being called in HVT. Sammon’s projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality while attempting to preserve the structure of inter-point distances in the projection.The centroids are plotted in 2D after performing Sammon’s projection at every level of the tessellation.

lets view the projected 2D centroids after performing sammon’s projection


hvt_torus_coordinates <-hvt.torus3[[2]][[1]][["1"]]
centroids <<- list()
  coordinates_value <- lapply(1:length(hvt_torus_coordinates), function(x){
    centroids <-hvt_torus_coordinates[[x]]
    coordinates <- centroids$pt
  })
centroid_coordinates<<- do.call(rbind.data.frame, coordinates_value)  
colnames(centroid_coordinates) <- c("x","y")
centroid_coordinates <- centroid_coordinates %>% data.frame() %>% round(4)
Table(head(centroid_coordinates), scroll = T, limit = 20)
x y
20.7876 8.8385
-15.0650 6.5840
4.1539 -7.6615
9.7266 -12.0612
-10.5474 18.8626
-6.0931 -12.1636

Step 3: Tessellation

The deldir package computes the Delaunay triangulation (and hence the Dirichlet or Voronoi tessellation) of a planar point set according to the second (iterative) algorithm of Lee and Schacter. For subsequent levels, transformation is performed on the 2D coordinates to get all the points within its parent tile. Tessellations are plotted using these transformed points as centroids.

plotHVT is the main function to plot hierarchical voronoi tessellations.

Now let’s try to understand plotHVT function. The parameters have been explained in detail below

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

For better visualisation, let’s plot the Voronoi tessellation.

muHVT::plotHVT(
  hvt.torus3,
  line.width = c(0.4),
  color.vec = c("#141B41"),
  centroid.size = 0.6,
  maxDepth = 1
)

From the presented plot, the inherent structure of the donut can be easily observed in the two-dimensional space

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data. .

Let’s have look at the function hvtHmap function which we will use to overlay a variable as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions

metric_list <- colnames(hvt.torus3[[3]]$summary)
metric_list <- metric_list[7:9]
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  hvt.torus3,
  torus_df,
  child.level = 1,
  hmap.cols = metric_list[[x]],
  line.width = c(0.4),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 1,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 900
)
})
grid.arrange(hmap[[1]], nrow = 1, ncol=1)

grid.arrange(hmap[[2]] ,nrow = 1, ncol=1)

grid.arrange(hmap[[3]], nrow = 1, ncol=1)

6.2 Refining Torus Projection at Level 2

In this section we will explore a little more and try to project the torus(3D) object to 2D Space with different set of model parameters at level 2 inorder to achieve more refined and detailed tessellation

Through this exploration, we can gain a deeper understanding of the transformation of the torus object into a 2D projection. By adjusting the model parameters specifically for the second level of tessellation, we expect to achieve a higher level of refinement and granularity in the resulting projection, enabling us to analyze the torus in greater detail and extract meaningful insights from the transformed data.

Step 1: Data Compression

First we will perform Hierarchical Vector Quantization using the torus data along with below mentioned model parameters to achieve a compression summary of atleast 80%.

for detailed information on data Compression please refer to section 2 of this vignette.

Model Parameters

set.seed(240)
hvt.torus4 <- muHVT::HVT(
  torus_df,
  n_cells = 20,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary for torus and see whether the model has achieved 80% compression or not.

compressionSummaryTable(hvt.torus4[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 20 0 0 n_cells: 20 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

It can be observed from the table above that none of the cells has hit the Quantization Error threshold. Therefore we can further subdivide the cells by increasing the n_cells inorder to achieve a compression of 80%

Since we are not able to achieve our compression objective. Let’s try again with the below mentioned set of model parameters.

Model Parameters

set.seed(240)
hvt.torus5 <- muHVT::HVT(
  torus_df,
  n_cells = 26,
  depth = 2,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Now,let’s checkout the compression summary for torus by.

compressionSummaryTable(hvt.torus5[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 26 0 0 n_cells: 26 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 666 535 0.8 n_cells: 26 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

It can be observed from the table above that only 535 cells out of 666 i.e. 80% of the cells hit the Quantization Error threshold. Since we have attained 80% compression we will not further subdivide the cells

Step 2: Data Projection

lets view the projected 2D centroids after performing sammon’s projection on the compressed data recieved after performing vector quantization.

For detailed information on Data Projection one can refer to section 3 of this vignette.


hvt_torus_coordinates <-hvt.torus5[[2]][[1]][["1"]]
centroids <<- list()
  coordinates_value <- lapply(1:length(hvt_torus_coordinates), function(x){
    centroids <-hvt_torus_coordinates[[x]]
    coordinates <- centroids$pt
  })
centroid_coordinates<<- do.call(rbind.data.frame, coordinates_value)  
colnames(centroid_coordinates) <- c("x","y")
centroid_coordinates <- centroid_coordinates %>% data.frame() %>% round(4)
Table(head(centroid_coordinates), scroll = T, limit = 20
             )
x y
20.2685 4.0434
-12.5504 10.9844
-2.1562 -5.5135
4.2848 -17.6823
-17.5999 11.8420
-13.2824 -10.8036

Step 3: Tessellation

Now, we have obtained the centroid coordinates resulting from the application of Sammon’s projection.

For better visualisation, let’s plot the Voronoi tessellation using the plotHVT function.

muHVT::plotHVT(
  hvt.torus5,
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  centroid.size = 0.8,
  maxDepth = 2
)

From the presented plot, the inherent structure of the donut can no longer be observed in the two-dimensional space

NOTE

When we pass the 3D torus data to the HVT function with depth 1, it performs a variance decomposition at a single level. This means that the algorithm calculates the overall variance of the data and provides a 2D visualization that represents the variance structure in a simplified manner, allowing us to visualize the data (donut structure).

However, when we increase the depth to 2, the HVT function performs a more detailed variance decomposition by breaking down the overall variance into subcomponents at two levels. This increased level of analysis introduces additional dimensions or factors that are not easily represented in a 2D visualization.

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data at level 2 for better visualization.

metric_list <- colnames(hvt.torus3[[3]]$summary)
metric_list <- metric_list[7:9]
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  hvt.torus5,
  torus_df,
  child.level = 2,
  hmap.cols = metric_list[[x]],
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 1,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 26
)
})
grid.arrange(hmap[[1]], nrow = 1, ncol=1)

grid.arrange(hmap[[2]] ,nrow = 1, ncol=1)

grid.arrange(hmap[[3]], nrow = 1, ncol=1)

7 Example Usage of Predictions using the predictLayerHVT on Personal Computers Dataset.

Data Understanding

In this section, we will use the Prices of Personal Computers dataset. This dataset contains 6259 observations and 10 features. The dataset observes the price from 1993 to 1995 of 486 personal computers in the US. The variables are price, speed, ram, screen, cd, etc. The dataset can be downloaded from here.

In this example, we will compress this dataset by using hierarchical VQ via k-means and visualize the Voronoi Tessellation plots using Sammons projection. Later on, we will overlay all the variables as a heatmap to generate further insights.

Here, we load the data and store into a variable computers.

set.seed(240)
# Load data from csv files
computers <- read.csv("https://raw.githubusercontent.com/Mu-Sigma/muHVT/master/vignettes/sample_dataset/Computers.csv")

Let’s explore the Personal Computers Dataset.

# Quick peek

Table(head(computers), scroll = T, limit = 20)
X price speed hd ram screen cd multi premium ads trend
1 1499 25 80 4 14 no no yes 94 1
2 1795 33 85 2 14 no no yes 94 1
3 1595 25 170 4 15 no no yes 94 1
4 1849 25 170 8 14 no no no 94 1
5 3295 33 340 16 14 no no yes 94 1
6 3695 66 340 16 14 no no yes 94 1

Now, let us check the structure of the data and analyse its summary.

str(computers)
#> 'data.frame':    6259 obs. of  11 variables:
#>  $ X      : int  1 2 3 4 5 6 7 8 9 10 ...
#>  $ price  : int  1499 1795 1595 1849 3295 3695 1720 1995 2225 2575 ...
#>  $ speed  : int  25 33 25 25 33 66 25 50 50 50 ...
#>  $ hd     : int  80 85 170 170 340 340 170 85 210 210 ...
#>  $ ram    : int  4 2 4 8 16 16 4 2 8 4 ...
#>  $ screen : int  14 14 15 14 14 14 14 14 14 15 ...
#>  $ cd     : chr  "no" "no" "no" "no" ...
#>  $ multi  : chr  "no" "no" "no" "no" ...
#>  $ premium: chr  "yes" "yes" "yes" "no" ...
#>  $ ads    : int  94 94 94 94 94 94 94 94 94 94 ...
#>  $ trend  : int  1 1 1 1 1 1 1 1 1 1 ...
summary(computers)
#>        X            price          speed              hd        
#>  Min.   :   1   Min.   : 949   Min.   : 25.00   Min.   :  80.0  
#>  1st Qu.:1566   1st Qu.:1794   1st Qu.: 33.00   1st Qu.: 214.0  
#>  Median :3130   Median :2144   Median : 50.00   Median : 340.0  
#>  Mean   :3130   Mean   :2220   Mean   : 52.01   Mean   : 416.6  
#>  3rd Qu.:4694   3rd Qu.:2595   3rd Qu.: 66.00   3rd Qu.: 528.0  
#>  Max.   :6259   Max.   :5399   Max.   :100.00   Max.   :2100.0  
#>       ram             screen           cd               multi          
#>  Min.   : 2.000   Min.   :14.00   Length:6259        Length:6259       
#>  1st Qu.: 4.000   1st Qu.:14.00   Class :character   Class :character  
#>  Median : 8.000   Median :14.00   Mode  :character   Mode  :character  
#>  Mean   : 8.287   Mean   :14.61                                        
#>  3rd Qu.: 8.000   3rd Qu.:15.00                                        
#>  Max.   :32.000   Max.   :17.00                                        
#>    premium               ads            trend      
#>  Length:6259        Min.   : 39.0   Min.   : 1.00  
#>  Class :character   1st Qu.:162.5   1st Qu.:10.00  
#>  Mode  :character   Median :246.0   Median :16.00  
#>                     Mean   :221.3   Mean   :15.93  
#>                     3rd Qu.:275.0   3rd Qu.:21.50  
#>                     Max.   :339.0   Max.   :35.00

Let us first split the data into train and test. We will use 80% of the data as train and remaining as test.

noOfPoints <- dim(computers)[1]
trainLength <- as.integer(noOfPoints * 0.8)
trainComputers <- computers[1:trainLength,]
testComputers <- computers[(trainLength+1):noOfPoints,]

K-means is not suitable for factor variables as the sample space for factor variables is discrete. A Euclidean distance function on such a space isn’t really meaningful. Hence, we will delete the factor variables(X, cd, multi, premium, trend) in our dataset.

Here we keep the original trainComputers and testComputers as we will use the variables from this dataset to overlay as heatmap and generate some insights.

trainComputers <-
  trainComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))
testComputers <-
  testComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))

Now, lets have a look at the scaled training dataset containing (5007 data points)

trainComputers <- scale(trainComputers) 

metric_list <- colnames(trainComputers)
scale_attr <- attributes(trainComputers)

trainComputers1 <- trainComputers %>% as.data.frame() %>% round(4)
Table(head(trainComputers1))
price speed hd ram screen ads
-1.2977 -1.1952 -1.3134 -0.7181 -0.6148 -2.3877
-0.7999 -0.7832 -1.2896 -1.1092 -0.6148 -2.3877
-1.1362 -1.1952 -0.8853 -0.7181 0.5490 -2.3877
-0.7091 -1.1952 -0.8853 0.0641 -0.6148 -2.3877
1.7228 -0.7832 -0.0766 1.6285 -0.6148 -2.3877
2.3956 0.9161 -0.0766 1.6285 -0.6148 -2.3877

Now, lets have a look at the scaled testing dataset containing (1252 data points).

testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) 
testComputers1 <- testComputers %>% as.data.frame() %>% round(4)
Table(head(testComputers1))
price speed hd ram screen ads
5008 -1.2287 -0.7832 -0.6760 -0.7181 0.5490 -0.8403
5009 1.3848 0.0922 3.0631 3.1928 0.5490 -0.8403
5010 -0.8016 0.0922 -0.6760 -0.7181 -0.6148 -0.8403
5011 0.2311 2.6668 -0.4096 -0.7181 -0.6148 -0.8403
5012 0.3084 0.9161 1.7311 1.6285 0.5490 -0.8403
5013 -0.5072 0.9161 3.0631 0.0641 -0.6148 -0.8403

7.1 Map A : Base Compressed Map

As we are familiar with the structure of the computers data, we will now follow the following steps to get the predictions using the Computers dataset.

Step 1: Data Compression

For more detailed information on Data Compression please refer to section 2 of this vignette.

Let us try to understand the HVT function first.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  projection.scale,
  normalize = T,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  diagnose = TRUE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of HVT function have been explained below :

First we will perform Hierarchical Vector Quantization using the computers dataset along with below mentioned model parameters to generate map A and try to achieve data compression of atleast 80%. In situations where the compression ratio does not meet the desired target we will iteratively increase the number of cells until the desired compression percentage is reached.

In this example usage we will compress the data at depth 1 as it provides a simplified overview of the data by calculating the overall variance and potentially reducing the dimensionality of the data

Model Parameters

set.seed(240)
hvt.results <- list()
map_A <- muHVT::HVT(trainComputers,   
                          n_cells = 1001,
                          depth = 1,
                          quant.err = 0.1,
                          projection.scale = 10,
                          normalize = F,
                          distance_metric = "L1_Norm",
                          error_metric = "max",
                          quant_method = "kmeans",
                          diagnose = F)

Now let’s check the compression summary. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

compressionSummaryTable(map_A[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 1001 831 0.83 n_cells: 1001 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error. Since we are successfully able to compress 83% of the data, so we will not further subdivide the cells

map_A[[3]] gives us detailed information about the hierarchical vector quantized data.

map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A

summaryTable(map_A[[3]]$summary)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 3 364 0.05 -0.46 0.92 -0.95 -0.72 -0.61 0.79
1 1 2 4 381 0.1 -1.02 0.92 -0.64 -0.72 0.55 -0.68
1 1 3 4 396 0.07 -0.40 0.92 -0.61 -0.72 -0.61 -0.57
1 1 4 5 660 0.07 0.04 0.92 0.17 0.06 0.55 0.93
1 1 5 2 591 0.01 -0.20 0.92 -0.50 0.06 0.55 0.08
1 1 6 4 741 0.05 1.28 -0.78 -0.41 -0.72 2.88 -0.38
1 1 7 7 720 0.06 0.02 0.92 0.85 0.06 0.55 1.52
1 1 8 5 905 0.08 1.82 0.92 0.68 1.63 0.55 0.28
1 1 9 5 594 0.03 -0.24 0.92 0.31 0.06 -0.61 0.07
1 1 10 2 391 0.02 -0.99 0.92 -0.08 -0.72 -0.61 1.01
1 1 11 5 190 0.06 -1.75 -0.78 -0.08 -0.72 -0.61 -0.08
1 1 12 3 68 0.08 -1.41 -1.20 -0.96 -0.72 0.55 0.83
1 1 13 6 87 0.05 -1.20 -1.20 -0.68 -0.72 -0.61 1.52
1 1 14 8 312 0.11 -0.72 0.09 -0.62 -0.72 -0.61 -0.71
1 1 15 13 987 0.16 1.49 0.09 3.06 3.19 0.55 -0.92
1 1 16 4 589 0.03 0.02 0.09 -0.08 0.06 0.55 0.72
1 1 17 8 647 0.1 0.50 0.09 -0.45 0.06 0.55 -1.67
1 1 18 2 37 0.04 -2.00 -0.78 -1.21 -1.11 -0.61 0.82
1 1 19 7 443 0.06 0.11 0.92 -0.64 -0.72 -0.61 -0.47
1 1 20 6 723 0.02 0.24 -0.78 0.82 1.63 -0.61 -0.30
1 1 21 5 102 0.07 0.32 -0.78 -0.81 -0.72 -0.61 -2.32
1 1 22 7 482 0.17 -0.04 2.67 -0.50 -0.72 -0.61 1.01
1 1 23 2 674 0.06 0.63 0.09 0.19 0.06 0.55 -1.08
1 1 24 7 820 0.09 0.86 0.09 0.09 0.06 2.88 0.36
1 1 25 8 714 0.14 1.15 0.92 -0.19 0.06 0.55 0.92
1 1 26 5 136 0.04 -1.53 -0.78 -0.69 -0.72 -0.61 -0.58
1 1 27 3 366 0.01 -0.71 0.92 -0.68 -0.72 -0.61 0.47
1 1 28 3 920 0.06 1.97 0.92 -0.08 0.06 2.88 0.72
1 1 29 3 182 0.01 -1.30 -0.78 -0.69 -0.72 -0.61 0.80
1 1 30 5 154 0.04 -0.26 0.92 -0.89 -0.72 -0.61 -2.27
1 1 31 3 360 0.08 -1.26 0.09 -0.23 -0.72 0.55 -0.45
1 1 32 7 457 0.05 -0.41 -1.20 0.33 0.06 -0.61 0.39
1 1 33 3 60 0.09 -1.82 -0.78 -0.50 -1.11 0.55 -0.16
1 1 34 8 468 0.08 -0.81 -0.78 0.23 0.06 0.55 0.45
1 1 35 6 187 0.02 -0.90 -0.78 -1.12 -0.72 -0.61 0.46
1 1 36 5 9 0.08 -0.60 -1.20 -0.85 -0.72 2.88 0.36
1 1 37 4 205 0.03 -1.26 -0.78 -0.65 -0.72 0.55 0.52
1 1 38 4 363 0.03 -0.68 0.92 -0.68 -0.72 -0.61 0.97
1 1 39 6 310 0.04 -0.65 0.09 -0.78 -0.72 -0.61 0.78
1 1 40 6 240 0.08 -1.41 -0.78 -0.08 -0.72 0.55 -0.51
1 1 41 2 55 0.04 -1.14 0.09 -1.18 -1.11 0.55 1.27
1 1 42 7 827 0.05 0.76 2.67 0.33 0.06 -0.61 1.52
1 1 43 3 556 0.02 -0.45 -0.78 0.82 0.06 0.55 0.07
1 1 44 9 194 0.03 -1.30 -0.78 -0.65 -0.72 -0.61 0.50
1 1 45 3 275 0.05 -1.03 0.09 -0.82 -0.72 -0.61 0.05
1 1 46 4 782 0.04 0.58 0.09 0.82 1.63 -0.61 1.01
1 1 47 9 217 0.07 -0.83 0.92 -1.18 -1.11 -0.61 0.91
1 1 48 2 988 0.05 2.89 0.92 3.06 1.63 -0.61 -1.37
1 1 49 7 506 0.04 -0.09 -0.78 0.33 0.06 -0.61 0.83
1 1 50 4 760 0.02 1.17 -0.78 0.46 1.63 -0.61 0.08
1 1 51 4 123 0.08 -1.35 -1.20 -0.75 -0.72 0.55 0.18
1 1 52 4 883 0.22 2.15 0.92 0.80 -0.13 0.55 -1.23
1 1 53 2 285 0.01 -0.97 -1.20 -0.68 0.06 -0.61 0.16
1 1 54 9 865 0.09 1.14 0.92 0.36 1.63 0.55 0.81
1 1 55 8 315 0.09 -1.16 0.92 -0.68 -0.72 -0.61 -0.73
1 1 56 2 608 0.05 -0.11 0.09 0.82 0.06 -0.61 1.27
1 1 57 5 610 0.08 0.35 0.09 0.43 0.06 -0.61 -0.44
1 1 58 1 799 0 0.38 0.09 0.82 1.63 -0.61 1.52
1 1 59 4 768 0.05 1.18 0.92 0.83 0.06 0.55 0.30
1 1 60 2 588 0.12 0.80 -0.78 -0.29 0.06 0.55 0.70
1 1 61 4 894 0.1 1.63 0.09 -0.08 0.06 2.88 0.92
1 1 62 5 168 0.05 -0.82 -0.78 -0.62 -0.72 -0.61 -1.30
1 1 63 8 878 0.1 1.13 0.92 0.07 0.06 2.88 0.37
1 1 64 2 912 0.01 1.34 1.38 0.82 1.63 -0.61 1.52
1 1 65 7 448 0.11 0.12 0.09 -0.08 -0.72 -0.61 0.61
1 1 66 3 896 0.13 1.04 0.92 -0.36 0.06 2.88 -0.87
1 1 67 2 717 0.01 0.46 -1.20 0.46 1.63 -0.61 0.08
1 1 68 8 155 0.08 -1.27 -0.78 -0.65 -0.72 0.55 -0.59
1 1 69 6 803 0.09 0.79 -0.92 0.46 1.63 -0.61 -1.67
1 1 70 7 622 0.1 0.97 0.92 -0.58 -0.72 0.55 0.39
1 1 71 3 328 0.04 -1.27 0.09 -0.08 -0.72 -0.61 0.44
1 1 72 6 505 0.04 -0.17 -0.78 0.33 0.06 -0.61 1.52
1 1 73 3 477 0.03 -0.43 -0.78 -0.08 0.06 0.55 0.91
1 1 74 3 696 0.01 -0.04 -0.78 0.82 1.63 -0.61 0.47
1 1 75 6 998 0.26 1.67 -0.07 3.06 3.19 2.88 -0.99
1 1 76 7 636 0.05 0.00 0.92 -0.08 0.06 0.55 0.28
1 1 77 5 344 0.04 -0.63 0.92 -0.68 -0.72 -0.61 1.52
1 1 78 3 895 0.01 0.13 -0.78 1.73 1.63 0.55 -1.30
1 1 79 5 755 0.04 0.14 -1.20 0.82 1.63 -0.61 1.52
1 1 80 2 367 0.01 0.22 -0.78 -0.69 0.06 -0.61 -2.23
1 1 81 3 138 0.02 -0.61 -0.78 -1.18 -1.11 -0.61 -0.44
1 1 82 3 558 0.03 0.38 0.09 -0.08 0.06 -0.61 0.08
1 1 83 7 642 0.07 -0.47 0.92 0.32 0.06 0.55 -0.46
1 1 84 5 667 0.04 -0.34 0.92 0.30 0.06 0.55 -1.30
1 1 85 3 983 0.07 1.52 0.92 3.06 3.19 0.55 0.08
1 1 86 4 885 0.07 2.73 0.92 0.81 0.06 0.55 0.10
1 1 87 6 162 0.04 -1.28 -0.78 -0.97 -0.72 -0.61 0.07
1 1 88 5 519 0.12 -0.88 0.92 0.16 -0.72 0.55 -0.56
1 1 89 8 848 0.19 -0.02 0.61 3.06 0.06 -0.61 -0.90
1 1 90 3 569 0.06 0.05 0.92 -0.23 0.06 -0.61 0.80
1 1 91 8 274 0.04 -0.48 -0.78 -0.65 -0.72 -0.61 0.90
1 1 92 2 511 0.06 -0.96 0.09 -0.08 0.06 0.55 0.08
1 1 93 1 713 0 -0.13 -1.20 0.82 1.63 -0.61 1.01
1 1 94 5 644 0.04 0.36 0.92 0.33 0.06 -0.61 0.81
1 1 95 4 626 0.09 -0.83 0.92 0.23 0.06 0.55 -0.59
1 1 96 2 5 0.09 -1.04 -0.78 -0.60 -0.91 2.88 -0.73
1 1 97 6 785 0.04 1.23 -0.78 0.82 1.63 -0.61 0.56
1 1 98 4 38 0.05 -1.71 -1.20 -1.18 -1.11 -0.61 0.83
1 1 99 3 614 0.04 0.33 0.09 -0.11 0.06 0.55 0.12
1 1 100 4 571 0.05 0.24 -0.78 0.86 0.06 -0.61 0.20
1 1 101 6 847 0.04 1.13 0.92 0.82 1.63 -0.61 0.37
1 1 102 5 211 0.01 -1.16 -0.78 -0.69 -0.72 -0.61 0.07
1 1 103 4 195 0.04 -0.42 -1.20 -1.12 -0.72 -0.61 0.56
1 1 104 5 227 0.02 -0.79 -0.78 -0.89 -0.72 -0.61 0.07
1 1 105 5 567 0.08 0.14 -0.78 0.82 0.06 -0.61 -0.55
1 1 106 9 765 0.11 0.17 2.67 0.26 0.06 -0.61 1.52
1 1 107 3 750 0.03 0.57 -0.78 0.84 1.63 -0.61 0.88
1 1 108 3 348 0.04 -0.94 0.09 -0.08 -0.72 -0.61 -0.56
1 1 109 5 612 0.04 0.56 0.09 0.33 0.06 -0.61 0.58
1 1 110 4 805 0.01 0.35 2.67 0.31 0.06 0.55 -0.30
1 1 111 4 797 0.1 0.20 -0.99 0.82 1.63 0.55 1.27
1 1 112 10 232 0.02 -0.94 -0.78 -0.68 -0.72 -0.61 0.43
1 1 113 3 921 0.01 0.97 2.67 0.82 1.63 -0.61 0.47
1 1 114 8 899 0.11 0.97 0.97 0.69 1.63 0.55 1.52
1 1 115 2 473 0.07 -0.92 0.92 -0.38 0.06 -0.61 -1.30
1 1 116 3 304 0.09 -0.84 0.09 -0.75 -0.72 0.55 -1.07
1 1 117 7 904 0.06 1.57 0.92 0.84 1.63 0.55 0.76
1 1 118 4 637 0.06 0.31 -0.78 0.89 0.06 0.55 0.45
1 1 119 3 116 0.04 -1.71 -1.20 -0.68 -0.72 -0.61 0.19
1 1 120 5 602 0.05 -0.53 0.92 -0.08 0.06 0.55 0.46
1 1 121 7 354 0.11 -0.09 0.09 -0.70 -0.72 -0.61 -1.08
1 1 122 7 456 0.04 -0.38 -1.20 0.33 0.06 -0.61 0.79
1 1 123 2 889 0.05 1.21 0.09 0.46 1.63 0.55 -1.37
1 1 124 3 963 0.01 1.17 -0.78 3.06 3.19 -0.61 0.47
1 1 125 4 494 0.06 -0.24 -1.20 0.45 0.06 -0.61 -0.44
1 1 126 3 84 0.01 -0.46 -0.78 -0.50 -0.72 -0.61 -2.35
1 1 127 4 65 0.06 -0.89 0.09 -1.18 -1.11 -0.61 -1.37
1 1 128 8 169 0.05 -0.74 -1.20 -1.09 -0.72 -0.61 0.55
1 1 129 4 438 0.07 -0.64 -1.20 0.45 0.06 -0.61 1.01
1 1 130 11 140 0.04 -1.11 -0.78 -0.67 -0.72 -0.61 1.52
1 1 131 8 743 0.04 0.62 -0.78 0.82 1.63 -0.61 0.45
1 1 132 4 855 0.11 1.17 0.09 0.73 1.63 0.55 -0.02
1 1 133 4 326 0.03 -0.89 -0.78 0.33 -0.72 -0.61 -0.33
1 1 134 7 322 0.05 -0.03 -0.78 -0.60 -0.72 -0.61 -0.44
1 1 135 1 133 0 -0.82 -1.20 -1.12 -0.72 -0.61 -0.44
1 1 136 7 418 0.1 0.10 0.92 -0.75 -0.72 -0.61 -1.25
1 1 137 5 655 0.04 -0.11 0.92 0.32 0.06 0.55 0.43
1 1 138 7 427 0.05 -0.41 -0.78 -0.08 0.06 -0.61 0.93
1 1 139 2 813 0.01 0.58 0.09 0.82 1.63 -0.61 1.52
1 1 140 4 176 0.02 -0.49 -0.78 -1.18 -1.11 -0.61 0.07
1 1 141 6 13 0.07 -0.97 -0.99 -0.89 -0.72 0.55 -2.35
1 1 142 3 752 0.02 0.60 -1.20 0.82 1.63 -0.61 0.58
1 1 143 5 570 0.1 0.07 0.92 -0.39 0.06 -0.61 -1.08
1 1 144 1 810 0 1.55 -0.78 0.82 1.63 -0.61 0.24
1 1 145 6 673 0.07 0.30 0.92 0.82 0.06 -0.61 -0.59
1 1 146 8 767 0.03 0.91 -0.78 0.82 1.63 -0.61 0.58
1 1 147 2 114 0.02 -0.81 -1.20 -1.18 -1.11 -0.61 0.37
1 1 148 8 196 0.03 -0.92 -1.20 -0.68 -0.72 -0.61 0.29
1 1 149 4 340 0.09 -1.30 -0.99 -0.08 0.06 -0.61 -0.11
1 1 150 7 759 0.08 0.63 -0.96 0.82 1.63 -0.61 -0.42
1 1 151 7 631 0.08 0.03 0.92 0.40 0.06 -0.61 1.01
1 1 152 5 479 0.05 -0.17 0.92 -0.08 -0.72 -0.61 0.07
1 1 153 2 449 0.04 -1.30 0.09 0.90 -0.72 0.55 -1.07
1 1 154 5 265 0.06 -1.09 -0.78 -0.08 -0.72 -0.61 -0.49
1 1 155 1 990 0 2.90 0.92 0.32 4.76 0.55 0.50
1 1 156 5 319 0.06 -1.03 -0.78 -0.08 -0.72 0.55 0.23
1 1 157 6 222 0.1 -0.96 0.09 -0.71 -0.72 -0.61 -1.27
1 1 158 7 706 0.08 0.20 0.92 0.86 0.06 0.55 -0.19
1 1 159 2 458 0.04 -0.63 0.09 0.30 -0.72 0.55 1.27
1 1 160 3 268 0.02 -0.83 -0.78 -0.08 -0.72 -0.61 1.52
1 1 161 9 331 0.08 -0.26 -0.78 -0.63 -0.72 0.55 0.80
1 1 162 4 440 0.06 0.06 0.09 -0.08 -0.72 -0.61 1.27
1 1 163 5 390 0.09 0.43 -0.87 -0.59 0.06 -0.61 -2.38
1 1 164 5 801 0.04 1.28 0.09 0.46 1.63 -0.61 0.08
1 1 165 7 977 0.27 1.37 2.67 0.68 1.63 2.88 0.55
1 1 166 3 225 0.08 -0.63 0.92 -1.15 -0.98 -0.61 -1.08
1 1 167 4 970 0.01 1.30 -0.78 3.06 3.19 -0.61 -0.30
1 1 168 3 902 0.15 1.72 -0.49 -0.08 1.63 -0.61 -2.39
1 1 169 4 103 0.03 -1.60 -1.20 -0.89 -0.72 -0.61 0.60
1 1 170 2 431 0.04 -0.97 0.09 -0.08 0.06 -0.61 1.27
1 1 171 7 19 0.19 -1.24 -0.96 -1.20 -1.05 0.55 -1.33
1 1 172 6 150 0.09 -0.38 -0.92 -0.68 -0.72 -0.61 -1.67
1 1 173 7 173 0.03 -0.68 -1.20 -1.12 -0.72 -0.61 0.16
1 1 174 7 422 0.11 -0.07 0.09 -0.67 -0.72 0.55 0.43
1 1 175 3 806 0.05 0.60 2.67 0.86 0.06 -0.61 0.47
1 1 176 9 826 0.14 0.58 0.92 -0.12 0.06 2.88 0.30
1 1 177 5 95 0.07 -0.98 -0.78 -1.18 -1.11 0.55 0.62
1 1 178 2 726 0.01 0.08 -1.20 0.82 1.63 -0.61 1.01
1 1 179 5 455 0.03 -0.48 -0.78 0.32 0.06 -0.61 1.52
1 1 180 6 224 0.09 -1.41 0.09 -0.65 -0.72 -0.61 -0.56
1 1 181 7 886 0.05 0.50 -0.78 1.73 1.63 0.55 -0.68
1 1 182 3 583 0.06 0.49 0.09 -0.11 0.06 -0.61 -1.08
1 1 183 3 897 0.01 0.33 -0.78 1.73 1.63 0.55 -1.30
1 1 184 5 851 0.1 1.04 0.09 0.83 1.63 0.55 0.63
1 1 185 5 26 0.09 -1.34 -1.20 -1.01 -0.87 -0.61 -1.67
1 1 186 6 2 0.1 -1.48 -1.13 -1.29 -1.11 -0.61 -2.29
1 1 187 3 462 0.03 0.37 0.92 -0.68 -0.72 -0.61 0.54
1 1 188 8 59 0.05 -1.52 -1.20 -1.18 -1.11 -0.61 0.40
1 1 189 6 428 0.11 0.86 -0.78 -0.24 -0.72 -0.61 0.66
1 1 190 1 719 0 -0.14 2.67 0.33 -0.72 0.55 -0.30
1 1 191 8 424 0.07 -0.64 0.92 -0.60 -0.72 0.55 0.69
1 1 192 6 352 0.04 -0.33 0.09 -0.68 -0.72 -0.61 0.41
1 1 193 2 828 0.01 0.84 0.09 0.82 1.63 -0.61 1.52
1 1 194 4 611 0.09 0.33 0.09 -0.18 0.06 0.55 0.75
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1 1 961 5 545 0.08 0.03 0.09 -0.62 0.06 0.55 0.30
1 1 962 4 423 0.07 -0.40 -0.78 -0.53 0.06 0.55 0.77
1 1 963 4 749 0.13 0.17 0.09 -0.08 0.06 2.88 -0.29
1 1 964 14 919 0.19 2.87 0.68 0.55 0.06 -0.61 -2.32
1 1 965 2 698 0.03 0.46 0.92 0.82 0.06 -0.61 -1.30
1 1 966 10 694 0.14 0.29 -0.78 -0.14 0.06 2.88 0.13
1 1 967 2 853 0.04 0.92 0.92 -0.69 -0.72 2.88 1.27
1 1 968 6 491 0.16 1.03 0.09 -0.47 -0.72 -0.61 0.45
1 1 969 5 692 0.12 0.18 1.01 0.17 0.06 0.55 1.52
1 1 970 6 110 0.07 -0.63 -0.78 -0.89 -0.72 -0.61 -1.67
1 1 971 5 946 0.04 0.98 2.67 1.73 1.63 -0.61 0.07
1 1 972 4 398 0.03 -0.23 0.92 -0.68 -0.72 -0.61 1.01
1 1 973 6 748 0.09 -0.22 -0.78 3.06 0.06 -0.61 -0.92
1 1 974 6 943 0.18 2.61 0.92 0.33 0.06 2.88 0.11
1 1 975 4 678 0.09 2.02 -0.78 0.46 0.06 -0.61 0.52
1 1 976 7 430 0.06 -0.01 0.92 -0.63 -0.72 -0.61 0.09
1 1 977 4 480 0.13 -0.53 0.09 -0.18 -0.72 2.88 0.27
1 1 978 4 373 0.1 -0.71 0.09 0.11 -0.72 -0.61 1.52
1 1 979 6 212 0.06 -1.23 -0.78 -0.69 -0.72 0.55 0.07
1 1 980 8 339 0.06 -0.05 -0.78 -0.56 0.06 -0.61 -2.30
1 1 981 3 736 0.02 0.35 -0.78 0.82 1.63 -0.61 1.01
1 1 982 10 357 0.11 -1.15 -0.95 -0.08 0.06 -0.61 0.54
1 1 983 4 764 0.14 0.12 -0.78 0.36 0.06 2.88 1.39
1 1 984 6 758 0.03 0.74 -0.78 0.84 1.63 -0.61 0.78
1 1 985 7 824 0.05 1.33 0.09 0.82 1.63 -0.61 0.57
1 1 986 4 208 0.02 -1.27 0.09 -1.08 -0.72 -0.61 0.06
1 1 987 11 313 0.03 -0.14 -0.78 -0.68 -0.72 -0.61 0.14
1 1 988 4 677 0.14 -0.34 0.92 -0.29 -0.72 2.88 0.08
1 1 989 8 729 0.18 0.16 2.67 0.27 0.06 -0.61 -0.43
1 1 990 6 263 0.05 -0.93 -0.78 -0.63 -0.72 0.55 0.39
1 1 991 2 148 0.04 -1.24 -0.78 -1.15 -0.72 -0.61 0.37
1 1 992 4 490 0.06 -0.47 0.92 0.31 -0.72 -0.61 -0.24
1 1 993 5 629 0.05 -0.41 0.92 0.81 0.06 -0.61 -0.49
1 1 994 6 12 0.17 -0.39 2.67 -1.02 -0.98 -0.61 1.52
1 1 995 20 492 0.09 0.17 0.09 -0.56 0.06 -0.61 -2.29
1 1 996 8 931 0.11 1.25 0.92 -0.53 0.06 2.88 -2.13
1 1 997 6 590 0.06 0.70 0.09 -0.08 0.06 -0.61 0.23
1 1 998 3 478 0.03 -0.35 -0.78 0.33 0.06 -0.61 0.18
1 1 999 3 762 0.07 1.05 0.92 -0.70 0.06 0.55 -2.28
1 1 1000 4 683 0.06 -0.60 0.92 0.87 0.06 0.55 -0.78
1 1 1001 4 47 0.02 -0.82 -0.78 -0.89 -0.72 -0.61 -2.25

Now let us understand what each column in the above summary table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Step 2: Data Projection

lets view the projected 2D centroids after performing sammon’s projection on the compressed data recieved after performing vector quantization.


hvt_torus_coordinates <-map_A[[2]][[1]][["1"]]
centroids <<- list()
  coordinates_value <- lapply(1:length(hvt_torus_coordinates), function(x){
    centroids <-hvt_torus_coordinates[[x]]
    coordinates <- centroids$pt
  })
centroid_coordinates<<- do.call(rbind.data.frame, coordinates_value)  
colnames(centroid_coordinates) <- c("x","y")
centroid_coordinates <- centroid_coordinates %>% data.frame() %>% round(4)
Table(head(centroid_coordinates))
x y
10.1594 -3.0833
9.0798 -11.7843
7.6534 -3.3278
-5.6131 -6.2331
-0.5750 -8.9096
-4.6926 -24.8100

Step 3: Tessellation

Now, we have obtained the centroid coordinates resulting from the application of Sammon’s projection.

For better visualisation, let’s plot the Voronoi tessellation for Map A using the plotHVT function.

# Voronoi tessellation plot for level one

 muHVT::plotHVT(map_A,
        line.width = c(0.2),  #
        color.vec = c("#141B41"),
        centroid.size = 0.01,  #1.5
        maxDepth = 1) 
Figure 3: The Voronoi Tessellation for layer 1 (map A) shown for the 1001 cells in the dataset ’computers’

Figure 3: The Voronoi Tessellation for layer 1 (map A) shown for the 1001 cells in the dataset ’computers’

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the computers dataset for better visualization.

metric_list <- colnames(trainComputers)
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = metric_list[[x]],
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.01,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 1001
)
})
grid.arrange(hmap[[1]], nrow = 1, ncol=1)

grid.arrange(hmap[[2]], nrow = 1, ncol=1)

grid.arrange(hmap[[3]], nrow = 1, ncol=1)

grid.arrange(hmap[[4]], nrow = 1, ncol=1)

grid.arrange(hmap[[5]], nrow = 1, ncol=1)

grid.arrange(hmap[[6]], nrow = 1, ncol=1)

7.2 Map B : Compressed Novelty Map

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

The identified_Novelty_cells along with the map A is passed to removeNovelty() function.

The output of removeNovelty() function is a list of two items: a dataset with novelty records and the subset of the dataset without novelty records.

library(plyr)
identified_Novelty_cells <<- c(213,384)
output_list <- removeNovelty(identified_Novelty_cells, map_A)
dataset_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

The datatable displayed below are the data with novelties.

colnames(dataset_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty%>% head(100) %>%   as.data.frame() %>%
Table(scroll = T, limit = 20)
Cell.ID Segment.Child price speed hd ram screen ads
188 213 -1.0521541 -0.7832055 -0.6759793 -0.718149 -0.6148117 -0.2979247
188 213 -1.1295176 -0.7832055 -0.6950076 -0.718149 -0.6148117 -0.2979247
188 213 -0.9529271 -0.7832055 -0.6759793 -0.718149 -0.6148117 -0.6169727
188 213 -1.1295176 -0.7832055 -0.6950076 -0.718149 -0.6148117 -0.6169727
188 213 -1.1446539 -0.7832055 -0.6759793 -0.718149 -0.6148117 -0.6169727
188 213 -1.0521541 -0.7832055 -0.6759793 -0.718149 -0.6148117 -0.6169727
188 213 -1.2051992 -0.7832055 -0.6759793 -0.718149 -0.6148117 -0.6169727
1000 384 2.4796551 0.9160675 8.2958340 1.628450 0.5490304 0.4677905
1000 384 2.0423835 0.9160675 8.2958340 1.628450 0.5490304 0.0689805

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) and highlights the identified outlier cell(s) in red on the map.

Let’s look at the Voronoi tessellation with the novelty cell(s) in the map highlighted in red.

plotCells(identified_Novelty_cells, map_A, line.width = c(0.2),centroid.size = 0.01 )
Figure 4: The Voronoi Tessellation with the novelty cells in the map highlighted in red

Figure 4: The Voronoi Tessellation with the novelty cells in the map highlighted in red

We pass the dataframe with novelty records to HVT function along with below mentioned model parameters to generate map B (layer 2).

Model Parameters

dataset_with_novelty <- dataset_with_novelty[,-1:-2]
map_B <- list()
map_B <- muHVT::HVT(dataset_with_novelty,
                  n_cells = 3,
                  depth = 1,
                  quant.err = 0.1,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F)

The datatable displayed below is the summary from map B

summaryTable(map_B[[3]]$summary)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 2 2 0.01 -1.09 -0.78 -0.69 -0.72 -0.61 -0.30
1 1 2 2 1 0.07 2.26 0.92 8.30 1.63 0.55 0.27
1 1 3 5 3 0.02 -1.10 -0.78 -0.68 -0.72 -0.61 -0.62

Now let’s check the compression summary. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

compressionSummaryTable(map_B[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 3 3 1 n_cells: 3 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

It can be observed from the table above that 3 cells out of 3 i.e. 100% of the cells has hit the Quantization Error threshold

7.3 Map C : Compressed Map without Novelty

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty and below mentioned model parameters.

Model Parameters

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 1001,
                  depth = 1,
                  quant.err = 0.1,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F)

The datatable displayed below is the summary from map C

summaryTable(map_C[[3]]$summary)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 5 952 0.09 0.79 -0.78 1.73 1.63 2.88 -0.94
1 1 2 6 187 0.04 -1.53 -0.78 -0.08 -0.72 -0.61 -0.69
1 1 3 9 655 0.1 0.14 0.92 0.06 0.06 0.55 0.86
1 1 4 2 589 0.01 -0.20 0.92 -0.50 0.06 0.55 0.08
1 1 5 1 24 0 0.18 -1.20 -0.69 -0.72 2.88 -0.30
1 1 6 7 719 0.06 0.02 0.92 0.85 0.06 0.55 1.52
1 1 7 4 897 0.06 1.58 0.92 0.83 1.63 0.55 0.14
1 1 8 3 972 0.01 1.47 0.92 3.06 3.19 -0.61 0.47
1 1 9 2 384 0.02 -0.99 0.92 -0.08 -0.72 -0.61 1.01
1 1 10 7 929 0.15 1.05 2.67 0.82 1.63 -0.61 -0.49
1 1 11 5 81 0.07 -1.31 -1.20 -0.98 -0.72 0.55 0.78
1 1 12 6 89 0.05 -1.20 -1.20 -0.68 -0.72 -0.61 1.52
1 1 13 5 292 0.07 -0.55 0.09 -0.73 -0.72 -0.61 -1.12
1 1 14 5 987 0.03 1.44 -0.78 3.06 3.19 0.55 -1.30
1 1 15 6 580 0.05 -0.03 0.09 -0.08 0.06 0.55 0.65
1 1 16 8 648 0.1 0.50 0.09 -0.45 0.06 0.55 -1.67
1 1 17 2 39 0.04 -2.00 -0.78 -1.21 -1.11 -0.61 0.82
1 1 18 7 434 0.06 0.11 0.92 -0.64 -0.72 -0.61 -0.47
1 1 19 7 561 0.06 -0.51 0.92 0.26 0.06 -0.61 -0.30
1 1 20 5 97 0.07 0.32 -0.78 -0.81 -0.72 -0.61 -2.32
1 1 21 7 693 0.06 0.07 0.92 0.87 0.06 0.55 0.52
1 1 22 2 676 0.06 0.63 0.09 0.19 0.06 0.55 -1.08
1 1 23 7 815 0.09 0.86 0.09 0.09 0.06 2.88 0.36
1 1 24 7 705 0.07 1.14 0.92 -0.21 0.06 0.55 0.83
1 1 25 2 128 0.03 -1.72 -0.78 -0.69 -0.72 -0.61 -0.37
1 1 26 3 358 0.05 -0.46 0.92 -0.95 -0.72 -0.61 0.79
1 1 27 3 237 0.01 -0.89 -0.78 -0.68 -0.72 -0.61 0.48
1 1 28 3 184 0.01 -1.30 -0.78 -0.69 -0.72 -0.61 0.80
1 1 29 5 107 0.04 -0.26 0.92 -0.89 -0.72 -0.61 -2.27
1 1 30 4 387 0.08 -1.13 0.09 -0.08 -0.72 0.55 0.24
1 1 31 7 459 0.05 -0.41 -1.20 0.33 0.06 -0.61 0.39
1 1 32 3 727 0.07 0.58 0.92 0.82 0.06 0.55 -0.22
1 1 33 4 445 0.05 -0.76 -0.78 -0.08 0.06 0.55 0.43
1 1 34 2 171 0.01 -1.32 -0.78 -0.89 -0.72 -0.61 0.56
1 1 35 5 8 0.1 -0.55 -1.20 -0.94 -0.72 2.88 0.63
1 1 36 4 19 0.09 0.15 -0.89 -1.12 -0.72 2.88 0.66
1 1 37 4 355 0.03 -0.68 0.92 -0.68 -0.72 -0.61 0.97
1 1 38 6 308 0.04 -0.65 0.09 -0.78 -0.72 -0.61 0.78
1 1 39 4 963 0.16 2.70 2.67 1.77 0.06 0.55 -0.54
1 1 40 2 51 0.04 -1.14 0.09 -1.18 -1.11 0.55 1.27
1 1 41 7 822 0.05 0.76 2.67 0.33 0.06 -0.61 1.52
1 1 42 9 421 0.07 -0.46 -0.78 -0.58 0.06 0.55 0.07
1 1 43 2 949 0.09 1.01 2.67 1.75 1.63 0.55 -0.39
1 1 44 3 271 0.05 -1.03 0.09 -0.82 -0.72 -0.61 0.05
1 1 45 4 778 0.04 0.58 0.09 0.82 1.63 -0.61 1.01
1 1 46 9 193 0.07 -0.83 0.92 -1.18 -1.11 -0.61 0.91
1 1 47 8 170 0.08 -1.27 -0.78 -0.65 -0.72 0.55 -0.59
1 1 48 7 501 0.04 -0.09 -0.78 0.33 0.06 -0.61 0.83
1 1 49 4 761 0.02 1.17 -0.78 0.46 1.63 -0.61 0.08
1 1 50 2 367 0.06 -1.47 0.92 -0.08 -0.72 -0.61 -0.27
1 1 51 4 878 0.22 2.15 0.92 0.80 -0.13 0.55 -1.23
1 1 52 2 285 0.01 -0.97 -1.20 -0.68 0.06 -0.61 0.16
1 1 53 10 869 0.06 1.34 0.92 0.40 1.63 0.55 0.61
1 1 54 4 702 0.05 0.02 0.92 0.82 0.06 0.55 -0.73
1 1 55 2 603 0.05 -0.11 0.09 0.82 0.06 -0.61 1.27
1 1 56 4 598 0.06 0.37 0.09 0.33 0.06 -0.61 -0.44
1 1 57 1 792 0 0.38 0.09 0.82 1.63 -0.61 1.52
1 1 58 4 763 0.05 1.18 0.92 0.83 0.06 0.55 0.30
1 1 59 1 646 0 1.23 -0.78 -0.08 0.06 0.55 0.77
1 1 60 7 378 0.08 -0.49 -0.78 0.38 -0.72 -0.61 0.48
1 1 61 5 990 0.06 1.52 0.92 3.06 3.19 0.55 -1.30
1 1 62 3 789 0.07 0.55 0.92 0.19 -0.72 2.88 0.46
1 1 63 2 910 0.01 1.34 1.38 0.82 1.63 -0.61 1.52
1 1 64 7 441 0.11 0.12 0.09 -0.08 -0.72 -0.61 0.61
1 1 65 3 889 0.13 1.04 0.92 -0.36 0.06 2.88 -0.87
1 1 66 2 714 0.01 0.46 -1.20 0.46 1.63 -0.61 0.08
1 1 67 6 991 0.07 2.13 0.92 3.06 3.19 0.55 -0.62
1 1 68 8 837 0.2 0.88 -0.89 0.33 1.63 0.55 -1.30
1 1 69 5 597 0.07 0.84 0.92 -0.60 -0.72 0.55 0.37
1 1 70 2 315 0.06 -1.21 0.09 -0.08 -0.72 -0.61 1.27
1 1 71 6 500 0.04 -0.17 -0.78 0.33 0.06 -0.61 1.52
1 1 72 4 494 0.08 -0.40 -0.78 0.03 0.06 0.55 0.94
1 1 73 9 182 0.06 -1.48 -0.78 -0.64 -0.72 -0.61 0.49
1 1 74 5 934 0.14 1.67 2.67 1.77 0.06 -0.61 -0.60
1 1 75 6 640 0.06 0.09 0.92 -0.08 0.06 0.55 0.28
1 1 76 5 343 0.04 -0.63 0.92 -0.68 -0.72 -0.61 1.52
1 1 77 2 892 0.01 0.17 -0.78 1.73 1.63 0.55 -1.30
1 1 78 5 752 0.04 0.14 -1.20 0.82 1.63 -0.61 1.52
1 1 79 2 374 0.01 0.22 -0.78 -0.69 0.06 -0.61 -2.23
1 1 80 1 247 0 -0.23 -0.78 -1.12 -0.72 -0.61 -0.44
1 1 81 6 585 0.06 0.70 0.09 -0.08 0.06 -0.61 0.23
1 1 82 2 557 0.02 -0.99 0.09 0.33 0.06 0.55 -0.84
1 1 83 12 613 0.09 -0.52 0.92 0.56 0.06 -0.61 -1.30
1 1 84 6 491 0.06 -0.79 -0.78 0.33 0.06 0.55 0.44
1 1 85 2 887 0.09 2.92 0.92 0.81 -0.33 0.55 0.04
1 1 86 9 207 0.04 -1.15 -0.78 -0.73 -0.72 -0.61 0.07
1 1 87 4 552 0.07 -0.20 -0.89 0.33 0.06 0.55 -0.41
1 1 88 3 686 0.06 0.29 0.92 0.82 0.06 -0.61 -1.30
1 1 89 3 560 0.06 0.05 0.92 -0.23 0.06 -0.61 0.80
1 1 90 8 279 0.03 -0.41 -0.78 -0.67 -0.72 -0.61 0.86
1 1 91 4 513 0.09 -0.39 -0.99 0.82 0.06 -0.61 0.95
1 1 92 4 798 0.1 0.20 -0.99 0.82 1.63 0.55 1.27
1 1 93 5 636 0.04 0.36 0.92 0.33 0.06 -0.61 0.81
1 1 94 3 385 0.09 0.21 -0.78 -0.55 -0.72 0.55 -1.15
1 1 95 2 844 0.02 0.24 -0.78 1.73 1.63 0.55 0.07
1 1 96 6 782 0.04 1.23 -0.78 0.82 1.63 -0.61 0.56
1 1 97 4 40 0.05 -1.71 -1.20 -1.18 -1.11 -0.61 0.83
1 1 98 4 532 0.09 0.38 0.09 -0.21 -0.72 0.55 0.70
1 1 99 5 565 0.06 0.18 -0.78 0.85 0.06 -0.61 0.18
1 1 100 6 859 0.04 1.41 0.92 0.82 1.63 -0.61 0.52
1 1 101 2 309 0.01 -1.57 0.09 -0.08 -0.72 -0.61 0.07
1 1 102 4 199 0.04 -0.42 -1.20 -1.12 -0.72 -0.61 0.56
1 1 103 3 224 0.01 -0.84 -0.78 -0.89 -0.72 -0.61 0.05
1 1 104 3 590 0.06 0.31 -0.78 0.82 0.06 -0.61 -0.56
1 1 105 9 757 0.11 0.17 2.67 0.26 0.06 -0.61 1.52
1 1 106 3 746 0.03 0.57 -0.78 0.84 1.63 -0.61 0.88
1 1 107 6 222 0.09 -1.41 0.09 -0.65 -0.72 -0.61 -0.56
1 1 108 10 607 0.06 0.54 0.09 0.33 0.06 -0.61 0.70
1 1 109 5 967 0.03 1.35 -0.78 3.06 3.19 -0.61 -0.30
1 1 110 17 794 0.16 0.60 -0.90 0.83 1.63 0.55 0.61
1 1 111 7 231 0.02 -0.96 -0.78 -0.68 -0.72 -0.61 0.41
1 1 112 6 925 0.06 1.06 2.67 0.82 1.63 -0.61 0.40
1 1 113 8 896 0.11 0.97 0.97 0.69 1.63 0.55 1.52
1 1 114 2 903 0 0.46 -0.78 1.73 1.63 0.55 -1.30
1 1 115 8 519 0.06 -0.50 -0.78 0.82 0.06 -0.61 -1.30
1 1 116 4 908 0.04 1.79 0.92 0.83 1.63 0.55 0.69
1 1 117 6 626 0.16 0.22 -0.78 0.89 -0.07 0.55 0.48
1 1 118 3 116 0.04 -1.71 -1.20 -0.68 -0.72 -0.61 0.19
1 1 119 7 780 0.11 0.03 0.92 0.10 0.06 2.88 0.43
1 1 120 5 369 0.09 0.04 0.09 -0.67 -0.72 -0.61 -1.08
1 1 121 7 458 0.04 -0.38 -1.20 0.33 0.06 -0.61 0.79
1 1 122 2 883 0.05 1.21 0.09 0.46 1.63 0.55 -1.37
1 1 123 3 867 0.06 1.13 -0.78 0.89 0.06 2.88 0.46
1 1 124 4 489 0.06 -0.13 -1.20 0.33 0.06 -0.61 -0.44
1 1 125 3 86 0.01 -0.46 -0.78 -0.50 -0.72 -0.61 -2.35
1 1 126 4 66 0.06 -0.89 0.09 -1.18 -1.11 -0.61 -1.37
1 1 127 8 175 0.05 -0.74 -1.20 -1.09 -0.72 -0.61 0.55
1 1 128 3 430 0.02 -0.66 -1.20 0.33 0.06 -0.61 1.01
1 1 129 11 142 0.04 -1.11 -0.78 -0.67 -0.72 -0.61 1.52
1 1 130 7 741 0.04 0.63 -0.78 0.82 1.63 -0.61 0.46
1 1 131 4 851 0.11 1.17 0.09 0.73 1.63 0.55 -0.02
1 1 132 5 240 0.12 -1.24 -0.78 0.44 -0.72 -0.61 -1.12
1 1 133 1 652 0 0.45 2.67 -0.41 -0.72 -0.61 -0.84
1 1 134 1 138 0 -0.82 -1.20 -1.12 -0.72 -0.61 -0.44
1 1 135 8 410 0.11 0.06 0.92 -0.72 -0.72 -0.61 -1.23
1 1 136 6 649 0.05 -0.16 0.92 0.32 0.06 0.55 0.42
1 1 137 9 425 0.06 -0.42 -0.78 -0.08 0.06 -0.61 0.87
1 1 138 2 811 0.01 0.58 0.09 0.82 1.63 -0.61 1.52
1 1 139 4 183 0.02 -0.49 -0.78 -1.18 -1.11 -0.61 0.07
1 1 140 6 15 0.07 -0.97 -0.99 -0.89 -0.72 0.55 -2.35
1 1 141 3 748 0.02 0.60 -1.20 0.82 1.63 -0.61 0.58
1 1 142 5 558 0.1 0.07 0.92 -0.39 0.06 -0.61 -1.08
1 1 143 1 809 0 1.55 -0.78 0.82 1.63 -0.61 0.24
1 1 144 4 350 0.05 -0.30 0.09 -0.68 -0.72 -0.61 -0.49
1 1 145 8 764 0.03 0.91 -0.78 0.82 1.63 -0.61 0.58
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1 1 911 5 336 0.08 -0.61 0.92 -1.18 -1.11 0.55 0.63
1 1 912 4 154 0.04 -1.39 -1.20 -0.61 -0.72 -0.61 0.36
1 1 913 4 528 0.05 0.22 0.09 -0.42 0.06 -0.61 -1.67
1 1 914 4 323 0.04 -0.80 0.92 -0.68 -0.72 -0.61 -1.30
1 1 915 2 596 0.01 0.17 0.09 0.33 0.06 -0.61 1.52
1 1 916 2 176 0.05 -1.10 -0.99 -1.18 0.06 -0.61 0.87
1 1 917 6 633 0.03 0.38 0.92 0.33 0.06 -0.61 0.37
1 1 918 5 366 0.1 -0.43 -0.78 -0.62 0.06 -0.61 -0.99
1 1 919 6 884 0.07 1.17 0.92 0.76 1.63 0.55 -0.37
1 1 920 5 994 0.39 1.72 0.92 3.29 2.25 2.88 -0.38
1 1 921 5 865 0.08 1.48 0.92 -0.07 1.63 0.55 0.73
1 1 922 3 849 0.01 1.60 0.92 -0.08 0.06 0.55 -2.35
1 1 923 7 710 0.09 1.29 -0.78 -0.41 -0.72 2.88 -0.19
1 1 924 6 390 0.1 0.18 0.09 -0.66 -0.72 -0.61 0.73
1 1 925 6 965 0.2 1.06 0.50 1.73 1.63 2.88 -0.92
1 1 926 2 936 0.01 1.55 2.67 0.82 1.63 -0.61 1.01
1 1 927 3 999 0.07 1.99 2.67 3.06 3.19 0.55 0.08
1 1 928 4 85 0.03 -1.37 -1.20 -1.12 -0.72 -0.61 -0.44
1 1 929 2 84 0.02 -2.01 -0.78 -0.50 -1.11 -0.61 0.07
1 1 930 3 252 0.02 0.05 -1.20 -1.12 -0.72 -0.61 0.50
1 1 931 4 637 0.03 -0.06 0.92 0.82 0.06 -0.61 0.07
1 1 932 2 985 0.05 2.89 0.92 3.06 1.63 -0.61 -1.37
1 1 933 13 12 0.08 -1.30 -1.20 -0.94 -0.72 -0.61 -2.29
1 1 934 8 725 0.12 0.72 0.92 -0.58 0.06 0.55 -2.27
1 1 935 3 708 0.1 1.56 0.92 0.07 -0.72 0.55 -0.39
1 1 936 4 462 0.07 -0.57 0.92 0.34 -0.72 -0.61 0.83
1 1 937 2 978 0.01 1.72 0.92 3.06 3.19 -0.61 0.47
1 1 938 4 272 0.02 -0.57 -0.78 -0.68 -0.72 -0.61 0.24
1 1 939 3 935 0.19 1.90 0.08 0.87 0.06 2.88 -1.08
1 1 940 6 470 0.02 -0.35 -0.78 0.33 0.06 -0.61 1.01
1 1 941 4 964 0.03 1.38 -0.78 3.06 3.19 -0.61 0.07
1 1 942 6 456 0.13 -0.63 0.09 -0.08 -0.72 2.88 -0.15
1 1 943 4 526 0.09 -0.41 -0.89 0.85 0.06 -0.61 1.52
1 1 944 3 17 0.1 -1.41 -1.06 -1.16 -0.98 0.55 1.52
1 1 945 7 855 0.07 2.44 0.92 0.79 0.06 0.55 0.17
1 1 946 3 125 0.02 -1.17 0.09 -1.18 -1.11 -0.61 0.77
1 1 947 9 198 0.03 -1.30 -0.78 -0.65 -0.72 -0.61 0.50
1 1 948 10 22 0.16 -0.20 2.67 -0.88 -0.87 -0.61 1.52
1 1 949 5 677 0.08 0.30 0.92 0.24 0.06 0.55 -0.38
1 1 950 7 539 0.05 -0.09 -0.78 0.83 0.06 -0.61 0.50
1 1 951 3 728 0.09 1.50 0.92 -0.55 0.06 -0.61 -2.15
1 1 952 4 512 0.1 0.03 0.92 -0.69 -0.72 0.55 -1.23
1 1 953 12 80 0.07 -0.46 0.09 -0.96 -0.72 -0.61 -2.29
1 1 954 9 659 0.04 0.68 0.92 0.33 0.06 -0.61 0.81
1 1 955 5 797 0.04 1.28 0.09 0.46 1.63 -0.61 0.08
1 1 956 2 713 0.01 0.12 -1.20 0.82 1.63 -0.61 0.37
1 1 957 7 233 0.01 -0.92 -0.78 -0.68 -0.72 -0.61 0.07
1 1 958 8 43 0.04 -0.89 -0.78 -0.89 -0.72 -0.61 -2.30
1 1 959 5 810 0.03 1.03 0.09 0.83 1.63 -0.61 0.76
1 1 960 7 345 0.11 -0.30 -0.78 -0.57 -0.72 0.55 0.16
1 1 961 3 173 0.09 -1.36 0.92 -0.81 -0.72 -0.61 -1.07
1 1 962 5 169 0.03 -0.83 0.09 -1.18 -1.11 -0.61 0.89
1 1 963 5 71 0.06 -1.21 -1.20 -0.98 -0.72 -0.61 -1.08
1 1 964 9 478 0.03 -0.31 -0.78 0.33 0.06 -0.61 0.43
1 1 965 1 129 0 0.89 -0.78 -0.60 -0.72 -0.61 -2.37
1 1 966 4 543 0.15 0.25 -0.78 -0.18 0.06 0.55 0.21
1 1 967 13 984 0.16 1.49 0.09 3.06 3.19 0.55 -0.92
1 1 968 8 117 0.03 -0.99 -0.78 -1.18 -1.11 -0.61 0.88
1 1 969 6 275 0.1 -0.66 -0.92 -0.65 -0.72 0.55 -0.45
1 1 970 1 843 0 0.55 0.92 0.82 1.63 -0.61 1.52
1 1 971 4 149 0.05 -1.55 -0.78 -0.61 -0.72 -0.61 1.01
1 1 972 5 541 0.08 0.03 0.09 -0.62 0.06 0.55 0.30
1 1 973 4 426 0.07 -0.40 -0.78 -0.53 0.06 0.55 0.77
1 1 974 4 738 0.13 0.17 0.09 -0.08 0.06 2.88 -0.29
1 1 975 14 920 0.19 2.87 0.68 0.55 0.06 -0.61 -2.32
1 1 976 4 163 0.05 -1.80 -0.78 -0.08 -0.72 -0.61 -0.46
1 1 977 9 681 0.11 0.27 -0.78 -0.19 0.06 2.88 0.12
1 1 978 5 821 0.15 1.09 0.92 -0.73 -0.72 2.88 0.83
1 1 979 6 476 0.16 1.03 0.09 -0.47 -0.72 -0.61 0.45
1 1 980 5 689 0.12 0.18 1.01 0.17 0.06 0.55 1.52
1 1 981 6 113 0.07 -0.63 -0.78 -0.89 -0.72 -0.61 -1.67
1 1 982 5 766 0.04 0.31 2.67 0.82 0.06 -0.61 0.07
1 1 983 4 394 0.03 -0.23 0.92 -0.68 -0.72 -0.61 1.01
1 1 984 7 974 0.04 1.04 -0.78 3.06 3.19 0.55 -0.81
1 1 985 6 938 0.18 2.61 0.92 0.33 0.06 2.88 0.11
1 1 986 4 672 0.09 2.02 -0.78 0.46 0.06 -0.61 0.52
1 1 987 5 422 0.04 -0.07 0.92 -0.61 -0.72 -0.61 0.07
1 1 988 3 395 0.06 -0.58 0.09 0.18 -0.72 -0.61 1.52
1 1 989 5 866 0.04 0.56 0.92 1.73 1.63 -0.61 0.07
1 1 990 8 348 0.06 -0.05 -0.78 -0.56 0.06 -0.61 -2.30
1 1 991 3 734 0.02 0.35 -0.78 0.82 1.63 -0.61 1.01
1 1 992 5 961 0.01 1.22 -0.78 3.06 3.19 -0.61 0.47
1 1 993 4 756 0.14 0.12 -0.78 0.36 0.06 2.88 1.39
1 1 994 6 755 0.03 0.74 -0.78 0.84 1.63 -0.61 0.78
1 1 995 7 975 0.27 1.37 2.67 0.68 1.63 2.88 0.55
1 1 996 4 205 0.02 -1.27 0.09 -1.08 -0.72 -0.61 0.06
1 1 997 6 320 0.03 -0.06 -0.78 -0.68 -0.72 -0.61 0.10
1 1 998 3 65 0.04 -0.56 -0.78 -0.08 -0.72 2.88 0.07
1 1 999 2 628 0.01 -0.21 0.92 0.82 0.06 -0.61 0.47
1 1 1000 5 218 0.05 -1.22 -0.78 -0.63 -0.72 0.55 0.51
1 1 1001 4 180 0.04 -1.01 -0.78 -1.12 -0.72 -0.61 0.44

Now let’s check the compression summary. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

compressionSummaryTable(map_C[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 1001 829 0.83 n_cells: 1001 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

It can be observed from the table above that 829 cells out of 1001 i.e. 83% of the cells has hit the Quantization Error threshold

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the computers dataset for better visualization.

metric_list <- colnames(trainComputers)
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  map_C,
  trainComputers,
  child.level = 1,
  hmap.cols = metric_list[[x]],
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.01,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 1001
)
})
grid.arrange(hmap[[1]],  nrow = 1, ncol=1)

grid.arrange(hmap[[2]],  nrow = 1, ncol=1)

grid.arrange(hmap[[3]],  nrow = 1, ncol=1)

grid.arrange(hmap[[4]],  nrow = 1, ncol=1)

grid.arrange(hmap[[5]],  nrow = 1, ncol=1)

grid.arrange(hmap[[6]],  nrow = 1, ncol=1)

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

Now, lets have a look at the scaled testing dataset containing (1252 data points).

#testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) 
testComputers1 <- testComputers %>% as.data.frame() %>% round(4)
Table(head(testComputers1))
price speed hd ram screen ads
5008 -1.2287 -0.7832 -0.6760 -0.7181 0.5490 -0.8403
5009 1.3848 0.0922 3.0631 3.1928 0.5490 -0.8403
5010 -0.8016 0.0922 -0.6760 -0.7181 -0.6148 -0.8403
5011 0.2311 2.6668 -0.4096 -0.7181 -0.6148 -0.8403
5012 0.3084 0.9161 1.7311 1.6285 0.5490 -0.8403
5013 -0.5072 0.9161 3.0631 0.0641 -0.6148 -0.8403

7.4 Prediction : predictLayerHVT

Now once we have built the model, let us try to predict using predictLayerHVT function and our test dataset which cell and which layer each point belongs to.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T,
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1,
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

Let’s see which cell and layer each point belongs to.

validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = F
  )
new_predict %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)
Row.Number Layer1.Cell.ID Layer2.Cell.ID
1 A155 C170
2 A987 C984
3 A312 C318
4 A663 C652
5 A913 C914
6 A848 C847
7 A987 C984
8 A521 C510
9 A433 C424
10 A145 C145
11 A976 C974
12 A503 C533
13 A903 C895
14 A955 C949
15 A621 C641
16 A903 C895
17 A312 C189
18 A621 C557
19 A224 C318
20 A989 C986

Now, lets understand the output of predictLayerHVT function.

8 Executive Summary

9 Applications

  1. Pricing Segmentation - The package can be used to discover groups of similar customers based on the customer spend pattern and understand price sensitivity of customers

  2. Market Segmentation - The package can be helpful in market segmentation where we have to identify micro and macro segments. The method used in this package can do both kinds of segmentation in one go

  3. Anomaly Detection - This method can help us categorize system behavior over time and help us find anomaly when there are changes in the system. For e.g. Finding fraudulent claims in healthcare insurance

  4. The package can help us understand the underlying structure of the data. Suppose we want to analyze a curved surface such as sphere or vase, we can approximate it by a lot of small low-order polygons in the form of tessellations using this package

  5. In biology, Voronoi diagrams are used to model a number of different biological structures, including cells and bone microarchitecture

  6. Using the base idea of Systems Dynamics, these diagrams can also be used to depict customer state changes over a period of time

10 References

  1. Topology Preserving Maps : https://link.springer.com/chapter/10.1007/1-84628-118-0_7

  2. Vector Quantization : https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-450-principles-of-digital-communications-i-fall-2006/lecture-notes/book_3.pdf

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : http://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : http://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation

  6. Embedding : https://en.wikipedia.org/wiki/Embedding